# Squares and Cubes

Students should learn the squares of numbers up to 32 and cubes upto 12 so that they do not waste time in he exam. Square roots of numbers up to 16 should also be learnt.

**Squares**

22 = 4 32 = 9 42=16 52=25 62=36 72=49 82=64 92=81

102 = 100 112 = 121 122 = 144 132 = 169 142 = 196 152 = 225 162 = 256 172 = 289

182 = 324 192 = 361 202 = 400 212 = 441 222 = 484 232 = 529 242 = 576 252 = 625

262 = 676 272 = 729 282 = 784 292 = 841 302 = 900 312 = 961 322 = 1024

**Square Roots**

√2 = 1.414 √3 = 1.732 √4=2 √5=2.236 √9=3

√10 = 3.162 √11 = 3.316 √12=3.464 √13 = 3.605

√14 = 3.741 √15 = 3.873

# Cubes

23=8 63=216 33=27 73=343 43=64 83=512 53 =125 93 =729

103 = 1000 113 = 1331 123 = 1728

*Note:**A number ending with 2, 3, 7 or 8 cannot be a perfect square. Only numbers ending with 0, 1, 4, 5, 6 and 9 can be perfect squares.*

**Illustration:**

The minimum of 3√2, 2√3, 2√5 and √11 is: 1. 3√2 2. 2√3 3. 2√5 4. √11

There are two ways of doing this sum.

The first way is to use the values of roots (as given above and memorised by the student) to get 3 x 1.41, 2 x 1.73, 2 ́ 2.23, and 3.31. We immediately see that the last value is the minimum.

The second method is to take the values under the root, as 3√2 = √18, 2√3 = √12, 2√5 = √20 and √11. Again we can see that √11 is the minimum. Both are easy methods.

Learning squares, cubes and square roots saves time in the exam. However, it is not necessary to calculate square roots of large numbers as such questions are seldom asked. Students tend to waste a lot of time finding tricks to find square roots, and our advice is to learn the roots upto the numbers given above. There is no need to find squares and roots of larger numbers.

If such a sum does appear, students can quickly figure out an approximation of the square root.

# Find the square root of 6084

Break up the number into pairs of 60 and 84.

What is the nearest square number less of equal to 60? We see that it is 49 and its root is 7.

So the first digit of the square root of 6084 is 7.

Now look at the second pair. Since it ends in 4, the second digit can be 2 or 8 only.Note that the square root of 70 is 4900 and that of 80 is 6400.

Since the number 6084 is nearer to 6400 than to 4900, we can immediately see that its square root is 78.

# Find the square root of 17956

Break up the number into pairs of 179 and 56. Using the above method. Fits we see what is the nearest square number to 179. We see that it is 169 and its root is 13. So the first two digits of the square root of the 17956 is 13.

Since the last digits are 56, the number can only end in 4 or 6.

The square of 130 is 16900 and that of 140 is 19600. Since 17956 is closer to 16900 than to 19600, the last digit is 4. So 134 is the square root of 17956.

The student need not waste time in finding roots of large numbers if one can master the above easy method.

**Powers**: Of late, sums with numbers raised to a large power have become popular with paper setters. An example of such a sum is given below:

**Illustration 4:**

What is the digit in the unit place in:

(264)102 + (264)103?

The tendency is to tick the choice with 8, since the sum of the last digits is 8, but this is wrong. The sum can easily be done if we break it up into factors:

264102[1 + 264] = 264102 [265]. We see now that the last digit must be 0, since an even number (which is the last digit of the first term) is multiplied by 5 (which is the last digit of the second term).

**Illustration 5:**

What is the last digit in 343?

Noticethatthelastdigitsformacycle.Listingthelastdigitsonly,weget31 =3,32 =9,33 =7,34 =1,35 =3,36 =9,...andsoon.Hence the digits 3, 9, 7, and 1 keep repeating after every 4th power. Then we have to divide 43 by 4, we get the remainder 3, hence the last digit must be 33 = 7.